We then proceed to renormalize the bare strong coupling constant \(\alpha _s^0\)  and the top quark mass \(m_t^0\)  in the \(\overline{\mathrm {MS}}\)  scheme. For this we require the strong coupling renormalization constant at two loops,
(since the leading-order contribution to the form factor does not depend on \(\alpha _s\) ) and the quark mass renormalization constant at three loopsBoth renormalization constants have been known for a long time. We take their
expressions from [1]}, where they are expressed in terms of \(\mathrm {SU}(N)\)  colour factors and are available as computer-readable files..
This renders the form factor finite, as there are no infrared divergences
present. This is in contrast to the gluon-gluon-Higgs form factor discussed in Ref. [2]}. Since the top quark does not appear as a dynamical degree of freedom at energy scales of the order of the Higgs boson mass, we
transform \(\alpha _s\)  from the six- to the five-flavour scheme, using the two-loop decoupling constantSince we express our results in terms of general \(\mathrm {SU}(N)\)  colour factors, we take the expressions from
[3]}, where they are available in a computer-readable form..
Furthermore, we also produce the form factor with the top quark mass renormalized in the on-shell scheme, which we obtain
from the \(\overline{\mathrm {MS}}\)  result by applying the \(\overline{\mathrm {MS}}\) -OS relation at three loops [4]}, [5]}, [6]}, [7]}.
